Exact-cylinder steady Stokes

Static walkthrough · canonical Marimo source available

Gallery

This page follows one frozen demonstration from exact authored geometry to a common Result. It does not execute a notebook or solve in the browser. The installed publication stays pinned to the exact producer revision. Source links and evidence dossiers instead resolve through the exact site build revision; none follows a branch.

Stage 1 🧭Problem setup

The fluid region is a $2.2\,\mathrm{m} \times 0.41\,\mathrm{m}$ channel with an exact authored circle centred at $[0.2, 0.2]\,\mathrm{m}$ and radius $0.05\,\mathrm{m}$. The problem is two-dimensional, steady, incompressible Stokes flow.

Exact authored geometry and boundary names. The later demonstration mesh approximates the circle with 50 straight chords; this diagram is explanatory, not geometric evidence.

The canonical Python composition authors that geometry explicitly:

python

geometry_graph = eqiora.geometry.CadAuthoredGraph.rectangle_extrusion(
    x_bounds=(0.0, 2.2),
    y_bounds=(0.0, 0.41),
    plane_z=0.0,
    depth=1.0,
    modeling_tolerance=1e-10,
).circular_through_cut(
    center=(0.2, 0.2),
    radius=0.05,
    boolean_tolerance=1e-10,
)
geometry = geometry_graph.planar_circular_section(
    classification_tolerance=1e-12,
    region="fluid",
    x_lower="inlet",
    x_upper="outlet",
    y_lower="walls",
    y_upper="walls",
    hole="cylinder",
)

Canonical Marimo geometry cell · exact geometry adapter · registered exact-geometry dossier · Python exact-geometry composition dossier

Stage 2 🧩Eqiora model definition

The accepted source spelling is easiest to read by first naming the Cauchy stress. With velocity $\boldsymbol{u}$, pressure $p$, dynamic viscosity $\mu$, identity tensor $\boldsymbol{I}$, and force potential $\phi$:

$$\boldsymbol{\sigma}(\boldsymbol{u},p) = 2\mu\,\operatorname{sym}(\nabla\boldsymbol{u}) - p\boldsymbol{I}$$

$$-\nabla\!\cdot\boldsymbol{\sigma}(\boldsymbol{u},p)-\nabla\phi=0, \qquad \nabla\!\cdot\boldsymbol{u}=0$$

Here $\mu=0.001\,\mathrm{kg}/(\mathrm{m}\,\mathrm{s})$, $\phi=0\,\mathrm{Pa}$, and pressure has source unit kg/(m*s^2), displayed as pascals. This is a notation projection of the Eqiora source below, not a new formulation.

Eqiora source form

text

relation momentum continuous on body {
  -div(
    2 * dynamic_viscosity * symmetric_part(grad(velocity))
    - isotropic_lift(pressure)
  ) - grad(force_potential) = 0;
}
relation incompressibility continuous on body {
  div(velocity) = 0;
}

Eqiora source form: packaged incompressible component · registered packaged incompressible source · registered packaged-Stokes dossier · registered direct Stokes model source

The application replays the packaged, byte-exact current Model rather than reconstructing scientific meaning in the notebook:

python

model_bytes = (
    files(eqiora)
    .joinpath("examples", "steady-flow-past-cylinder.model.json")
    .read_bytes()
)
model = eqiora.replay(model_bytes)

Eqiora source form: canonical Marimo replay cell · current Model identity dossier

Stage 3 🔷Mesh and boundaries

The inlet profile uses $U_{\max}=0.3\,\mathrm{m}/\mathrm{s}$ and channel height $H=0.41\,\mathrm{m}$:

$$g(y)=\frac{4U_{\max}y(H-y)}{H^2}, \qquad \boldsymbol{u}_{\mathrm{in}}=[g(y),0]$$

At the outlet, $\operatorname{normal}(\boldsymbol{\sigma})=0$. The walls and cylinder use $\boldsymbol{u}=0$.

Eqiora source form

text

relation inlet_profile_definition continuous on body {
  inlet_profile
    - 4 * inlet_speed * coordinate(1) * (channel_height - coordinate(1))
      / channel_height ^ 2 = 0;
}
relation inlet_velocity continuous on x_lower {
  trace(velocity) + normal(isotropic_lift(inlet_profile)) = 0;
}
relation outlet_traction continuous on x_upper {
  normal(
    2 * dynamic_viscosity * symmetric_part(grad(velocity))
    - isotropic_lift(pressure)
  ) = 0;
}
relation lower_wall continuous on y_lower { trace(velocity) = 0; }
relation upper_wall continuous on y_upper { trace(velocity) = 0; }

Accepted registered-case source form for the exact-hole closure

text

walls_and_cylinder: trace(u) = [0, 0] m/s

Eqiora source form: application boundary relations · accepted walls-and-cylinder trace boundary

The bounded realization requests at most 50 circular-boundary facets, then resolves and generates the source-bound mesh:

python

mesh_request = eqiora.meshing.MeshRequest(
    maximum_boundary_error=1e-4,
    minimum_mean_ratio=1e-5,
    maximum_boundary_facets=50,
)
mesh_plan = eqiora.meshing.resolve(geometry, mesh_request)
mesh = eqiora.meshing.generate(geometry, plan=mesh_plan)

Deliberately coarse demonstration mesh

The accepted realization has 50 straight chords and 104 affine triangles. All 104 mesh vertices lie on a boundary; 103 velocity vertices are essential, and the only free velocity vertex is the outlet midpoint $[2.2,0.2]\,\mathrm{m}$. This walkthrough makes no curved-element, mesh-convergence, or PDE-convergence claim.

Chordal reference-mesh dossier · chordal realization-binding dossier · canonical Marimo mesh cell · plain Python mesh adapter · Python mesh composition dossier

Stage 4 ▶Submit and result

The immutable SteadyStokes intent makes every scale and solver bound explicit. fluid.resolve binds the accepted Model and Mesh into a SteadyStokesPlan; the application then creates one Run with one eqiora.submit call and obtains one common Result with run.result().

python

stokes_intent = eqiora.fluid.SteadyStokes(
    length_scale_m=0.41,
    velocity_scale_m_per_s=0.3,
    pressure_scale_pa=0.001 * 0.3 / 0.41,
    relative_tolerance=1e-6,
    absolute_tolerance=1e-13,
    maximum_iterations=10_000,
)
stokes_plan = eqiora.fluid.resolve(model, stokes_intent, mesh=mesh)

run = eqiora.submit(model, plan=stokes_plan)
result = run.result()

Eqiora source form: canonical intent/submit/result cells

The real carriers in that composition are Geometry, MeshPlan, Mesh, Model, SteadyStokesPlan, Run, Result, and SteadyStokesEvidence. The admitted Result lineage binds the following public identities:

Carrier boundary	Public identity
Model	Result.model_digest
exact Geometry	Result.mesh(FieldRef).source_digest
authored correspondence	Result.mesh(FieldRef).correspondence_digest
Mesh	Result.mesh(FieldRef).digest
SteadyStokesPlan realization	Result.run_manifest().realization_digest
Run / Result	Result.run_manifest().digest
pressure FieldSnapshot	Result.field(FieldRef).digest

These names expose existing identity links; the page does not invent another aggregate digest or pretend that static documentation executed the run.

Registered Plan-and-Run dossier

Stage 5 🌡Pressure visualization

The canonical plotting cell selects the pressure FieldSnapshot from the common Result and passes it to the general scalar-field adapter:

python

pressure = result.snapshots[0]
evidence = eqiora.fluid.steady_stokes_evidence(result)
pressure_figure = eqplot.plot_scalar_field(result, field=pressure.field)

Presentation, not evidence.Units: PaPressure (Pa), frozen exact-cylinder steady-Stokes demonstration at c6b7a21f52ae1acf941d26319d2499ed89152c15; presentation only, not validation.Open canonical plotting cell

Result evidence · Pressure-still presentation case

Read the figure only as a pressure field in pascals, with a viridis scale and the 104-triangle affine mesh overlaid. The admitted pixels are a static presentation of the linked Result; color and pixel values are not validation and do not establish flow quality.

Stage 6 ✅Verified and not claimed

Verified boundary

Supported

• One frozen 2D steady incompressible Stokes exact-cylinder demonstration, rendered from its accepted public Result path and linked evidence.
• The exact geometry, source-bound chordal realization, packaged model, Plan-to-Run composition, and Marimo application each have a directly linked registered dossier.
• The installed publication record binds the admitted PNG, exact source revision, Result lineage, caption, alternative text, and presentation-only boundary.

Not claimed

• No curved elements.
• No mesh/PDE convergence.
• No drag/lift coefficient, scaled or mesh-independent force, or DFG value.
• No transient or Navier–Stokes behavior.
• No vortex shedding.
• No 3D.
• No production mesher.
• No performance claim.
• No cross-platform/byte-reproducible result.
• No pixel validation.
• API presence is neither verification nor maturity.

Browse the evidence catalog · exact geometry · chordal mesh · steady-Stokes science · common Result path · Marimo composition

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